Multigrid Computation of Maxwell Eigenvalues

We consider the problem of solving the discrete Maxwell eigenvalue problem , w > 0, in a closed simply connected cavity Ω ⊂ R³. For related eigenvalue problems for symmetric second order elliptic operators, efficient iterative schemes for the computation of a couple of the smallest eigenvalues/eigenvectors have been proposed [1]. They are based on a preconditioned inverse iteration and a comprehensive analysis has been presented in [3].

In the case of the Maxwell eigenvalue problem the large kernel of the curl-operator thwarts the straightforward application of these algorithms. However, when the discretization is based on edge elements, we have an explicit representation of Kern(curl) through gradients of linear finite element functions.

This paves the way for a fast approximate projection onto Kern(curl)^⊥, which can be coupled with the edge element multigrid scheme developed by the author [2]. Numerical experiments confirm the good performance of this approach for large scale problems.

References

1. J. BRAMBLE, A. KNYAZEV, AND J. PASCIAK, A subspace preconditioning algorithm for eigenvector/eigenvalue computation, Advances Comp. Math., 6 (1996), pp. 159-189.
2. R. HIPTMAIR, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal., 36 (1999), pp. 204-225.
3. K. NEYMEYR, A geometric theory for preconditioned inverse iteration applied to a subspace, Tech. Rep. 130, SFB 382, Universität Tübingen, Tübingen, Germany, November 1999. Submitted to Math. Comp.